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In writing my senior thesis I met the following problem: Sometimes I have some intuition about some mathematical statement. Yet I find it extremely painful trying to put these intuition into precise form on paper. In particular it is very hard to specify the correct condition for statement.

Does anyone have some tips for me in doing so? How do you often do it?

Let me elaborate a bit further. I believe it is an experience that any mature mathematician must have went through. We want to go to D, and we need to go through A, B, C. But A can not be stated clearly until one knows B, B can not be stated clearly until one knows A and C, and C can not be stated clearly until one knows B. But we sort of have a vague picture of A, B, C in our mind. It sounds very stupid, but I don't know where to start.

I wonder if this question is too vague for MO. So please close it if you see fit.

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I guess I will try some strategies suggested by everyone. –  Tran Chieu Minh Apr 13 '10 at 11:48
I run into this issue a lot when I am writing something up. Occasionally it really is just that my thinking is a little bit disorganized, and I have to work at it for awhile until everything falls into place. But I have found that more often than not this issue is actually caused by substantive mathematical considerations. It generally isn't that A, B, and C are simply wrong and can't be made precise; it's usually that some deeper statements D and E clarify A, B, and C. Sometimes I can identify D and E on my own, but the best thing is really to ask someone (as others have suggested). –  Paul Siegel Apr 13 '10 at 12:09
I guess that is what math is all about: making an intuition precise. Very good summary of what mathematicans try to do! –  vonjd Apr 13 '10 at 12:38
I think it is interesting question, but perhaps it belongs in a place more suited for discussion. –  S. Carnahan Apr 13 '10 at 14:10
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closed as not a real question by Harry Gindi, Qiaochu Yuan, Kevin H. Lin, Mariano Suárez-Alvarez, S. Carnahan Apr 13 '10 at 14:05

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5 Answers

I see this in students all the time, and I always give the same advice: talk to somebody. Find a friend who would be willing to listen and challenge you on every point. Sit down with a piece of paper, and try to tell her/him the whole story, explain the theorem you are trying to prove, examples, counter-examples, etc. Even if your friend can't help you formalize the ideas, the act of explaining itself might prove very useful to clarify the matter. Good luck!

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The only possible modification I would make to this advice is to change "a piece of paper" to "a whiteboard or blackboard". I find it is easier to erase and modify things there than on paper. But this is a nit -- the high order bit is to talk. –  Neel Krishnaswami Apr 13 '10 at 7:07
Just wondering: what if the issue one is dealing with is not within the fields of interest of his colleagues and friends, hence he can't discuss it with them? How to find someone for discussion of the idea, without feeling scared that he might profit (via stealing the idea if it shows to be good) from it, and leave you empty handed? –  Harun Šiljak Apr 13 '10 at 7:39
This fear of somebody "stealing the idea" is very much misdirected, I think. It is much more useful to fear missing something obvious, some old textbook reference, some connections to other problems, etc. In the event your friend or colleague helps you with your problem or offers some new idea, not only you don't loose anything, you make a math progress and might gain a co-author. That's a good thing... –  Igor Pak Apr 13 '10 at 8:15
That was, more or less, the thought I had in the first place - some old-school scholars convinced me in possibilities of 'intellectual theft', but I stood by the claim that there is more to gain than to lose - and MO seems like a fine place to look for answers - even if you don't want to ask it in full, you ask for portions of the answer, meet people who are into the area, and make the necessary advance. –  Harun Šiljak Apr 13 '10 at 8:22
Not to be too self-promotory, but we had a very long discussion about this at the SBS a few years ago sbseminar.wordpress.com/2008/07/19/followup-working-in-secret –  Ben Webster Apr 13 '10 at 13:44
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My first intuition is always to talk to somebody. This can help, since the speech center is a different part of the brain, works differently, taps different parts of the brain. In that way, even speaking out loud what you want to talk about can supply you with a surge of inspiration. Don't forget that written words are just a way to store the sounds we communicate with.

That said a second opinion is always very good. On the one hand someone smarter might have a ready answer, but on the other hand explaining the material (maybe from ground up, if that's not too much) to someone who doesn't know anything about it yet is apt can give you a lot of inspiration.

Try thinking about the object at hand in terms of real-life words. Ask yourself 'what happens if it does X? What if it does Y?' trying to learn a bit more about the object at hand - just like a physicist could pick up a mechanical contraption, look at it from many angles, and try turning a gear here and there, we have to do that with mathematical objects. Most of this is done on paper, using formulas and words.

Sometimes you can't put together theorems because the notation you came up with isn't intuitive enough - try thinking of a system of notation that limited to your current point of interest will be coherent, complete, and intuitive.

However sometimes some things simply will not 'tick' unless seen or heard: in signal processing you sometimes must listen to (some form of) the signals or functions at hand to understand what's going on. Fourier analysis helps visualize, but it doesn't do justice to the information you can get. In real analysis the picture of a saddle point cannot be replaced by any amount of writing - you just have to see it. Similarly an animation cannot be replaced with a few pictures. Example: I would have never noticed this effect had the display not been animated MO: effect in additive resynthesis

Try visualizing your mathematical objects. Ask a fellow geometer to help you figure out a pretty illustration. Visualizing is important because, again, it taps into different parts of our brain, all which can work for the cause rather than sitting around doing nothing. And they work better when they're looking at something pretty, rather than ugly! If you visualized the data/theorem already, try visualizing differently. Maybe you're thinking of some specific representative of the kind of object your theorem talks about, while the theorem visualizes better on a different one? (example: intermediate value theorems don't visualize too well on straight line functions..)

Another option is it might be a "writer's block". Just forget about it for some time, and get back to it later. Stop thinking about it, have a full day without working on maths, meditate, take a nap, go to the movies, sleep over it, have a good, long, enjoyable, dinner, relax listening to some music.

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In addition to the other answers...experiment! Write SAGE (or other) code to look at a hundred or a thousand examples of what you're trying to say something about. You'll probably see the pattern more clearly when it's sitting right in front of you in numerical form, and probably catch a class of exceptions you hadn't thought of before. If your particular intuition isn't easily calculable on a computer, then generate as many examples as you can by hand, and organize the data in such a way that your brain can see any patterns more clearly (and use SAGE or other code to simplify tedious calculations along the way).

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Could you explain what SAGE code is? –  Rasmus Bentmann Apr 16 '10 at 14:38
Sage: Open Source Mathematics Software. I can't speak highly enough about it. Link: sagemath.org –  Cam McLeman Apr 16 '10 at 17:28
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You think about it and try to clarify your ideas, till you can write them up precisely. If there is another way, I am not aware of it.

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I would only add that the act of writing up your ideas is often itself very clarifying, even if what you end up with bears little resemblance to what you thought you were going to write up. –  Pete L. Clark Apr 13 '10 at 6:27
Yes, absolutely. Thinking about mathematics also involves writing discussing it with others. Examples are extremely important: start from the stupidest, most degenerated case you can think of, analyze it thoroughly, then move to slightly less stupid cases, and so on. –  Angelo Apr 13 '10 at 6:48
Sounds like Feynman's problem solving algorithm! –  Kevin H. Lin Apr 13 '10 at 8:01
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I would say the "correct conditions" for a statement are discovered by trying to prove it and seeing what you need to be true in order for the statement to follow.

Proofs are generally discovered in the opposite direction than they are written because proofs are written for elegance and conciseness, not for teaching purposes. The last thing you discover is the first step of your elegantly-presented proof. Which is why textbooks can be very confusing to students. They understand the statement of the theorem and then the proof starts with "Let blah blah be " and the student is left wondering where that came from. Well, it came from spending a long time working on the proof... and the process of discovery is almost never shown.

Sometimes the very statement of a theorem is confusing because it is not clear where they got the maddeningly detailed conditions of the theorem. They got them by starting with the conclusion and working out the proof.

And sometimes the basic definitions of an entire theory are obscure and baroque, and only much later one realises that the definition is just what one needs for a certain big theorem to be true.

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As an aftethought: you might want to get ahold of Polya's book "How to Solve it". Googling about, I have also found a book called "How to Prove it" by Daniel J. Velleman. The former is a classic. Does anyone have an opinion of the latter? –  Miguel Apr 13 '10 at 11:07
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